## Input-output conditions for the asymptotic behavior of linear skew-product flows and applications.(English)Zbl 1142.47025

Let $$\sigma$$ be a discrete flow on a metric space $$\Theta$$. Let $$X$$ be a Banach space; denote by $${\mathcal L}(X)$$ the Banach algebra of bounded linear operators on $$X$$. A pair $$\pi=(\Phi,\sigma)$$ is called a discrete linear skew-product flow on $$X\times\Theta$$ if the mapping $$\Phi:\Theta\times{\mathbb N}\to{\mathcal L}(X)$$ has the properties (i) $$\Phi(\theta,0)$$ is the identity mapping and (ii) $$\Phi(\theta,m+n)=\Phi(\sigma(\theta,n),m)\Phi(\theta,n)$$. Let $$\Delta({\mathbb Z},X)$$ be the space of sequences $$s:{\mathbb Z}\to X$$ for which the sets $$\{k\in{\mathbb Z}: s(k)\neq 0\}$$ are finite. The authors give conditions under which $$\pi$$ is exponentially dichotomic in terms of properties of solutions of the equation $\gamma(n+1)=\Phi(\sigma(\theta,n),1)\gamma(n)+s(n+1),$ where $$\theta\in\Theta$$ and $$s\in\Delta({\mathbb Z},X)$$.

### MSC:

 47D06 One-parameter semigroups and linear evolution equations 34D09 Dichotomy, trichotomy of solutions to ordinary differential equations 93D25 Input-output approaches in control theory 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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